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Unformatted text preview: 15-1 Chapter 15: Statistical Thermodynamics Problem numbers in italics indicate that the solution is included in the Students Solutions Manual.Questions on Concepts Q15.1)What is the relationship between ensemble energy and the thermodynamic concept of internal energy? The ensemble energy is equal to the difference in internal energy at some finite temperature to that present at 0 K. Q15.2)List the energetic degrees of freedom expected to contribute to the internal energy at 298 K for a diatomic molecule. Given this list, what spectroscopic information do you need to numerically determine the internal energy? Translational, rotational, and vibrational energetic degrees of freedom are expected to contribute to the internal energy at 298 K. Translational degrees of freedom will be in the high-temperature limit; therefore, a molar contribution of 1/2RT per degree of freedom is expected. Rotations are also expected to be in the high-temperature limit. Since a diatomic has two non-vanishing moments of intertia, a molar contribution of RT is expected. Finally, one needs to know the vibrational frequency to determine if the high-temperature limit is appropriate, and if not to determine the vibrational contribution to the internal energy. Q15.3)List the energetic degrees of freedom for which the contribution to the internal energy determined by statistical mechanics is equal to the prediction of the equipartition theorem at 298 K. The high-temperature approximation will generally be applicable to translations and rotations at 298 K; therefore, these degrees of freedom will make contributions to the internal energy in accord with the equipartition theorem. Q15.4)Write down the contribution to the constant volume heat capacity from translations and rotations for an ideal monatomic, diatomic, and nonlinear polyatomic gas, assuming that the high-temperature limit is appropriate for the rotational degrees of freedom. monatomic diatomic Non-linear polyatomic translations 3/2 R 3/2 R 3/2 R rotations 0 R 3/2 R Chapter 15/Statistical Thermodynamics 15-2 Q15.5)When are rotational degrees of freedom expected to contribute Ror 3/2R(linear and nonlinear, respectively) to the molar constant volume heat capacity? When will a vibrational degree of freedom contribute Rto the molar heat capacity? The rotational degrees of freedom will contribute R or 3/2R to the molar constant-volume heat capacity when the high-temperature limit is valid, defined as when T > 10Rwhere Ris the rotational temperature, defined as B/k. A vibrational degree of freedom will contribute R to the molar heat capacity when the high-temperature limit is applicable, defined as when T > 10Vwhere Vis the vibrational temperature, defined as /k....
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